91影视

We have an M/M/1 queueing system with a limited waiting room of size 2. The parameters of the system are: 1. Arrival rate (位): 3 customers per hour 2. Service rate (碌): 4 customers per hour (since they have an average service time of \(\frac{1}{4}\) hour) 3. Maximum number of customers in the system (c): 2

The traffic intensity (蟻) is given by the ratio of the arrival rate to the service rate, i.e., \(\rho = \frac{\lambda}{\mu}\). Plugging in 位=3 and 碌=4, we get: \[ \rho = \frac{3}{4} \]

The probability of zero customers in the system (P鈧�) is given by the following formula: \[ P_0 = \frac{1}{1+\rho+\rho^2} \] Plugging in 蟻=\(\frac{3}{4}\), we get: \[ P_0 = \frac{1}{1+(\frac{3}{4})+(\frac{3}{4})^2} \approx 0.5714\]

The average number of customers in the system (L) is given by the following formula: \[ L = \rho + \frac{\rho^2(2-\rho)}{2(1-\rho^2)}\] Plugging in 蟻=\(\frac{3}{4}\), we get: \[ L = \frac{3}{4} + \frac{(\frac{3}{4})^2(2-\frac{3}{4})}{2(1-(\frac{3}{4})^2)} \approx 1.1429 \] The average number of customers in the shop is approximately 1.1429.

The probability of the shop being full, i.e., having two customers in the system is given by P鈧�. \[P_2 = P_0 \cdot \rho^2\] Plugging in P鈧�\(\approx 0.5714\) and 蟻=\(\frac{3}{4}\), we get: \[ P_2 \approx 0.5714 \cdot (\frac{3}{4})^2 \approx 0.2143 \] The proportion of potential customers that enter the shop is 1 - P鈧�, which is approximately \( 1-0.2143 = 0.7857 \).

If the barber works twice as fast, the service rate will be doubled, i.e., 碌' = 8 customers per hour. Recalculate the traffic intensity (蟻'), probability of zero customers (P鈧�'), average number of customers in the shop (L'), and the proportion of potential customers that enter the shop. 1. New traffic intensity, 蟻' = \(\frac{\lambda}{\mu'} = \frac{3}{8}\) 2. New probability of zero customers, P鈧�' = \[\frac{1}{1+\frac{3}{8}+(\frac{3}{8})^2} \approx 0.6912\] 3. New average number of customers, L' = \[\frac{3}{8} + \frac{(\frac{3}{8})^2(2-\frac{3}{8})}{2(1-(\frac{3}{8})^2)} \approx 0.9926 \] 4. New proportion of potential customers that enter the shop, 1 - P鈧�' = \[1 - 0.6912 \cdot (\frac{3}{8})^2 \approx 0.9661 \] The difference in the average number of customers in the shop is L' - L: \[ 0.9926 - 1.1429 = -0.1503 \approx -0.150\] Doubling the barber's speed won't significantly affect the number of customers in the shop, as the difference is approximately -0.150; instead, it will improve the proportion of potential customers that enter the shop (from 0.7857 to 0.9661).